The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that the trace of A is 3 and the trace of A 3 is - 18 then the value of the determinant of A is_____
A. A
Let A = a b c 3 - a     ⇒   A = 3 a - a 2 - b c A 3 = a b c 3 - a a b c 3 - a a b c 3 - a = a 2 + b c a b + 3 b - a b a c + 3 c - a c c b + ( 3 - a ) 2 a b c 3 - a = a 2 + b c 3 b 3 c c b + ( 3 - a ) 2 a b c 3 - a tr A 3 = a 3 + a b c + 3 b c + 3 b c + 3 b c - a b c + ( 3 - a ) 3 = - 18 a 3 + 9 b c + 3 - a 3 = - 18 9 b c + 27 - 9 a ( 3 - a ) = - 18 b c + 3 - 3 a + a 2 = - 2 3 a - a 2 - b c = 5    ⇒    | A | = 5
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